# SECTION 4.2# Example 1 de := diff(a*y(t),t,t) + b*diff(y(t),t) + c*y(t) = 0;

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subs(y(t) = exp(r*t),de);

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expand(%);

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%/exp(r*t);

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simplify(%);

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subs({a=1,b=5,c=-6},%);

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solve(%,r);

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# Example 2solve(r^2+2*r-1,r);# Solve auxilliary equation.

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rsol := %;

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y1 := exp(rsol[1]*t);

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y2 := exp(rsol[2]*t);

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y := c1*y1 + c2*y2;

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subs(t=0,y);

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eq1 := % = 0;

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diff(y,t);

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subs(t=0,%);

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eq2 := % = -1;

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solve({eq1,eq2},{c1,c2});

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subs(%,y);

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# Solution using dsolve:y := 'y';

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de := diff(y(t),t,t) + 2*diff(y(t),t) - y(t) = 0;

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dsolve(de);

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dsolve({de,y(0)=0,D(y)(0)=-1});

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# Example 3de := diff(y(t),t,t) + 4*diff(y(t),t) + 4*y(t) = 0;

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y1 := exp(-2*t);

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y2 := t*exp(-2*t);

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subs(y(t)=y1,de); expand(%);

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subs(y(t)=y2,de); expand(%);

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y := c1*y1 + c2*y2;

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subs(t=0,y);

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eq1 := % = 1;

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diff(y,t);

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subs(t=0,%);

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eq2 := % = 3;

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solve({eq1,eq2},{c1,c2});

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subs(%,y);

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# Solution using dsolve:y := 'y';

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de;

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dsolve(de);

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dsolve({de,y(0)=1,D(y)(0)=3});

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# Example 4de := diff(y(t),t$3) + 3*diff(y(t),t$2) - diff(y(t),t) - 3*y(t) = 0;

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P := r^3 + 3*r^2 - r -3 = 0;

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factor(P);

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solve(P,r);

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rsol := %;

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y1 := exp(rsol[1]*t);

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y2 := exp(rsol[2]*t);

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y3 := exp(rsol[3]*t);

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y := c1*y1 + c2*y2 + c3*y3;

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# Note that this solution is equivalent to the one in the text since c1, c2, c3 are arbitrary constants.# Solution using dsolve:y := 'y';

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de;

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dsolve(de);

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